Quasiballistic Transport for Discrete One-Dimensional Quasiperidic Schr\"odinger Operators

TL;DR

This paper establishes power-law lower bounds on quantum transport for one-dimensional quasiperiodic Schr"odinger operators, revealing quasiballistic behavior under certain frequency conditions.

Contribution

It provides a quantitative ballistic lower bound for the time evolution of periodic Schr"odinger operators, extending understanding of quasiballistic transport in quasiperiodic systems.

Findings

Power-law lower bounds on transport for quasiperiodic operators

Uniform bounds across all p>0

Conditions on frequencies for quasiballistic behavior

Abstract

We obtain (up to logarithmic scaling) the power-law lower bound $M_{p}(T_{k})\gtrsim T_{k}^{(1-\delta)p}$ on a subsequence $T_{k}\rightarrow\infty$, uniformly across $p>0$, for discrete one-dimensional quasiperiodic Schr\"odinger operators with frequencies satisfying $\beta(\alpha)>\frac{3}{\delta}\min_{\sigma}\gamma$. We achieve this by obtaining a quantitative ballistic lower bound for the Abel-averaged time evolution of general periodic Schr\"odinger operators in terms of the bandwidths. A similar result without uniformity, which assumes $\beta(\alpha)>\frac{C}{\delta}\min_{\sigma}\gamma$, was obtained earlier by Jitomirskaya and Zhang, for an implicit constant $C<\infty$.